{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 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0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 261 "" 0 "" {TEXT 256 19 " Dynamics Revision:" }} {PARA 259 "" 0 "" {TEXT 271 15 "The Phase Plane" }{TEXT -1 0 "" }}} {EXCHG {PARA 256 "" 0 "" {TEXT 257 23 "(Version 2.0, 21/02/00)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "Th is revision exercise is concerned with exploring the phase plane for t he " }{TEXT 258 15 "simple pendulum" }{TEXT -1 51 ", which is describe d by the differential equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 257 "" 0 "" {TEXT 261 1 "x" }{TEXT -1 2 "'(" }{TEXT 262 1 "t" }{TEXT -1 4 ") = " }{TEXT 263 1 "y" }{TEXT -1 1 "(" }{TEXT 264 1 "t" }{TEXT -1 3 "), " }{TEXT 265 1 "y" }{TEXT -1 2 "'(" }{TEXT 266 1 "t" }{TEXT -1 9 ") = -sin(" }{TEXT 267 1 "x" }{TEXT -1 1 "(" }{TEXT 268 1 "t" }{TEXT -1 2 "))" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "(note that we've set " }{TEXT 269 1 "m" } {TEXT -1 7 "=1 and " }{TEXT 270 1 "g" }{TEXT -1 32 "=1 for simplicity) , and for the " }{TEXT 259 16 "Duffing equation" }{TEXT -1 1 "," }}} {PARA 258 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 260 "" 0 "" {TEXT 272 1 "x" }{TEXT -1 2 "'(" }{TEXT 273 1 "t" }{TEXT -1 4 ") = " }{TEXT 274 1 "y" }{TEXT -1 1 "(" }{TEXT 275 1 "t" }{TEXT -1 3 "), " }{TEXT 276 1 "y" }{TEXT -1 2 "'(" }{TEXT 277 1 "t" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "x(t)-x(t)^3;" "6#,&-%\"xG6#%\"tG\"\"\"*$-F%6#F'\"\"$!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 16 "Setting up Maple" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 87 "Run the following commands to load in the extra function packages used in this session." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "with(plots);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "(Remember that you can use \"" }{TEXT 281 1 ":" }{TEXT -1 15 "\" in place of \"" }{TEXT 282 1 ";" }{TEXT -1 32 "\" to suppress unwanted outputs.)" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 35 "1. The phase plane for th e pendulum" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "First, we define the differential equations and initial c onditions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "pendEqns:= di ff(x(t),t)=y(t), diff(y(t),t)=-sin(x(t));" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "pendInits:= x(0)=3.1,y(0)=0.0;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "We \+ run " }{TEXT 283 6 "dsolve" }{TEXT -1 23 " to generate a solution" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "sol1:= dsolve(\{pendEqns,pen dInits\},\{x(t),y(t)\},numeric);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "And then use " } {TEXT 284 7 "odeplot" }{TEXT -1 55 " (from the \"plots\" package) to p lot this orbit in the (" }{TEXT 279 1 "x" }{TEXT -1 2 ", " }{TEXT 280 1 "y" }{TEXT -1 13 ") phase plane" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "odeplot(sol1,[x(t),y(t)],0..20,numpoints=75);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Note that if you name plots (with \"" }{TEXT 278 2 ":= " }{TEXT -1 51 "\") you can combine them on the same axes using the " }{TEXT 285 7 "display" }{TEXT -1 24 " command -- for example:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "pendInits1:= x(0)=1.5,y(0)=0 .0;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "dsolv e(\{pendEqns,pendInits1\},\{x(t),y(t)\},numeric);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "p1:=odeplot(%,[x(t),y(t)],0. .10,numpoints=75):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "pendI nits2:= x(0)=0.5,y(0)=0.0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "dsolve(\{pendEqns,pendInits2\},\{x(t),y(t)\},numeric);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "p2:= odeplot(%,[x(t) ,y(t)],0..7,numpoints=75):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "display([p1,p2],scaling=constrained);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "(The \"sc aling=constrained\" option specifies that the axes should have equal s cales.)" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 10 "Exercise 1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 100 "Plot a SINGLE picture of the phase plane showing the orbits having the following initial conditions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " (i) x(0) = 0. 5, y(0) = 0.0;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 " (ii) x(0) = 1.5, y(0) = 0.0;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 " (iii) x(0) = 3.1, y(0) = 0.0; " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 " (iv) x(0) = -3.5, y(0) = 0.5;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 " (v) x(0) = +3.5, y(0) = -0.5;" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 " (vi) x(0) = -3.5, y(0 ) = 1.0." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 182 "You will need to vary the time-range (i.e. as in \"0..7 \" above) to get (at least) one complete orbit in each case. You may a lso need to vary \"numpoints\" to get a smooth-looking curve." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Yo u should also insert some textual notes into your Maple document to re mind yourself about" }}{PARA 0 "" 0 "" {TEXT -1 109 "the different kin ds of orbits -- i.e. oscillating, rotating, and the separatrix that se parates the two types." }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 0 "" }} {PARA 3 "" 0 "" {TEXT -1 43 "2. The phase plane for the Duffing equati on" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The Duffing equation can be set up in Maple as follows:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "duffEqn:=\{diff(x(t),t)=y(t) ,diff(y(t),t)=x(t)-x(t)^3\};" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "duffInits1:= x(0)=0.5,y(0)=0.0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "duffInits2:= x(0)=sqrt(2),y(0)=0.01;" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 53 "There is a special command in the \"DEtools\" package, \+ " }{TEXT 286 6 "DEplot" }{TEXT -1 176 ", which allows you to construct phase plane pictures more quickly -- you have to specify the differen tial equations, the time range, and a list of different initial condit ions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "DEplot(duffEqn, [ x(t),y(t)],0..35,[[duffInits1],[duffInits2]],\n arrows=NONE,stepsize=0 .1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 10 "Exercise 2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 170 "Complete the picture of the phase plane for the Duffing equation, showing also a periodic solution insi de the left-hand \"lobe\", and some solutions outside the separatrix. " }}}}{MARK "0 0 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 }